Project 1¶
Part 1: Prospective and orthographic projections¶
Question 1¶
Perspective¶
Orthographic¶
Histogram equalization stretched the intensity distribution from a narrow range to cover more of 0–255. Shadows and highlights separated more clearly, increasing local contrast. In this image, the equalized version appears [cleaner / more detailed in dark regions / slightly harsh in highlights]. Overall, I think the quality [improved / improved moderately / over-enhanced noise in flat areas], which matches the histogram spreading we see.
The naive method (Gaussian → derivative) and the DoG method should look nearly identical in both x and y directions.
Small differences can appear due to:
Discretization of the derivative (finite differences vs the continuous derivative),
Kernel size/truncation (we use ±3σ support),
Boundary handling (we used reflect padding),
Slight normalization differences between [-1,0,1] and the DoG scaling.
We implemented unsharp masking by smoothing with a Gaussian (σ = 1.2), subtracting to obtain the high-frequency component, and adding it back with weights α ∈ {1,2,4,8,10}. For both CT and moon images, moderate α (≈2–4) improved perceived sharpness without obvious artifacts. At high α (≥8), sharpening introduced halos along strong edges and amplified background noise, confirming the trade-off between acutance and artifact visibility. Adjusting σ shifts which structures are enhanced: smaller σ emphasizes fine texture/noise, while larger σ targets broader edges.
Smaller K protects more edges (less diffusion near gradients) but denoises less; larger K smooths more broadly and can round edges. More iterations/Δt ⇒ stronger smoothing.
Gaussian smoothing reduces noise but blurs across edges, softening boundaries. Perona–Malik diffusivity drops near large gradients, so it denoises within regions while preserving/Sharpening edges (piecewise-smooth look). Differences also depend on parameters (σ vs K, iterations, Δt) and boundary handling.
Limitations of anisotropic diffusion (Perona–Malik)
Parameter sensitivity: Results hinge on 𝐾 K, Δ 𝑡 Δt, and the iteration count; small changes can under/over-smooth and there’s no built-in stopping rule.
Texture loss / cartooning: Fine textures and low-contrast edges are often treated as noise and flattened (piecewise-constant look).
Edge leakage & shift: If gradients are weak relative to 𝐾 K, diffusion can still cross edges, softening or shifting boundaries.
Discretization issues: 4-neighbour schemes aren’t perfectly rotation-invariant; explicit updates require small Δ 𝑡 Δt (slow) and can create boundary artifacts.
Noise model mismatch: Works best for roughly Gaussian noise; speckle/impulse noise is handled poorly.
One improvement: Structure-tensor–guided (coherence-enhancing) diffusion
Instead of using a scalar conductance based only on ∣ ∇ 𝐼 ∣ ∣∇I∣, estimate local orientation with the structure tensor 𝐽
𝐺 𝜌 ∗ ( ∇ 𝐼 𝜎 ∇ 𝐼 𝜎 ⊤ ) J ρ
=G ρ
∗(∇I σ
∇I σ ⊤
). Diagonalize 𝐽 𝜌 J ρ
to get dominant directions; then diffuse strongly along coherent directions (within edges/lines) and weakly across them via a diffusion tensor
𝑅 d i a g ( 𝛼 , 𝛽 ) 𝑅 ⊤ D=Rdiag(α,β)R ⊤ with 𝛼 ≫ 𝛽 α≫β. This preserves thin/elongated structures and reduces edge blurring/leakage compared with PM.
Why it helps: PM only knows “how strong is the gradient?”; coherence-enhancing diffusion also knows “which way does the structure run?”, so it smooths noise in the safe direction while protecting edge normal directions—better on textures, ridges, vessels, wood grain, etc.